Systems and methods for deblurring data corrupted by shift variant blurring

ABSTRACT

The present invention provides methods, systems and machine readable medium including machine readable code for deblurring data corrupted by shift variant blurring. A first version of data having shift variant blurring characterized by a first shift variant point spread function is provided. A target shift invariant point spread function is selected. A second shift variant point spread function is derived wherein a combination of the first and second shift variant point spread functions generates the target shift invariant point spread function. The second shift variant point spread function is applied to the first version of the data thereby generating a second version of the data having shift invariant blurring characterized by the target shift invariant point spread function. A linear shift invariant filter is applied to the second version of the data thereby generating a deblurred version of the data.

RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/621,555, filed on Jan. 9, 2007, which is hereby incorporated byreference.

GOVERNMENT LICENSE RIGHTS

This invention was made with government support under grant numbersEB001489 and EB003298 awarded by National Institutes of Health. TheGovernment has certain rights to this invention.

FIELD

The invention generally relates to data processing and more particularlyto systems and methods for deblurring data corrupted by shift variantblurring.

BACKGROUND

Imaging devices are routinely used in the medical industry. Examples ofsuch imaging devices include, but are not limited to, Positron EmissionTomography (PET) imaging devices and Single Photon Emission ComputedTomography (SPECT) imaging devices. The imaging devices capture twodimensional projection images of a target or a patient. The twodimensional projection images are reconstructed into a three dimensionalvolumetric image of the target or the patient. The reconstructed imagesare often corrupted by uniform attenuation, non-uniform attenuation, andshift variant blurring.

One prior art method for compensating for the shift variant blurringinvolves the use of a frequency-distance principle (FDP) algorithm. TheFIR algorithm is a pre-processing algorithm. In other words, the FDPalgorithm attempts to compensate for shift variant blurring in the twodimensional projection images of the target prior to using the twodimensional projection images to reconstruct the three dimensional imageof the target. The FDP algorithm makes an assumption that there is nouniform or non-uniform attenuation present in the two dimensionalprojection images generated by the imaging devices. This assumptionoften results in the amplification of noise and the introduction ofartifacts in the reconstructed three dimensional image of the target.

Another prior art method for compensating for shift variant blurringinvolves the use of an intrinsic iterative reconstruction algorithm. Theintrinsic iterative reconstruction algorithm is implemented during thereconstruction of the two dimensional projection images of the targetinto a three dimensional image of the target. While the intrinsiciterative reconstruction algorithm is effective at compensating forshift variant blurring, the implementation of the intrinsic iterativereconstruction algorithm can take hours or even days. Such longprocessing times to obtain a three dimensional image of a target canlead to delays in diagnosis and treatment of a patient's medicalcondition.

Another prior art method involves the use of shift invariant filters toattempt to compensate for shift variant blurring. Shift invariantfilters are typically not very effective at filtering shift variantblurring.

Thus what is needed is a system and method for deblurring data corruptedby shift variant blurring to overcome one or more of the challengesand/or obstacles described above.

SUMMARY

One aspect of the invention provides a method of deblurring datacorrupted by shift variant blurring. The method includes providing afirst version of data having shift variant blurring characterized by afirst shift variant point spread function, selecting a target shiftinvariant point spread function, deriving a second shift variant pointspread function wherein a combination of the first and second shiftvariant point spread functions generates the target shift invariantpoint spread function, applying the second shift variant point spreadfunction to the first version of the data thereby generating a secondversion of the data having shift invariant blurring characterized by thetarget shift invariant point spread function, and applying a linearshift invariant filter to the second version of the data therebygenerating a deblurred version of the data.

Another aspect of the invention provides a machine readable medium forstoring a machine executable program for deblurring data corrupted byshift variant blurring. The machine readable medium includes machinereadable code for providing a first version of data having shift variantblurring characterized by a first shift variant point spread function,machine readable code for selecting a target shift invariant pointspread function, machine readable code for deriving a second shiftvariant point spread function wherein a combination of the first andsecond shift variant point spread functions generates the target shiftinvariant point spread function, machine readable code for applying thesecond shift variant point spread function to the first version of thedata thereby generating a second version of the data having shiftinvariant blurring characterized by the target shift invariant pointspread function, and machine readable code for applying a linear shiftinvariant filter to the second version of the data thereby generating adeblurred version of the data.

Another aspect of the invention provides a method for deblurring datacorrupted by shift variant blurring. The method includes providing afirst version of data having shift variant blurring, blurring the firstversion of the data further using a shift variant blurring kernelthereby generating a second version of data having shift invariantblurring, and applying a linear shift invariant filter to the secondversion of the data thereby generating a deblurred version of the data.

The foregoing and other features and advantages of the invention willbecome further apparent from the following detailed description of thepresently preferred embodiments, read in conjunction with theaccompanying drawings. The detailed description and drawings are merelyillustrative of the invention rather than limiting the scope of theinvention being defined by the appended claims and equivalents thereof.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example and not limitedin scope to the accompanying figures, in which like reference numeralsindicate similar elements, and in which:

FIG. 1 is a perspective view of one example of an embodiment of anuclear imaging assembly in accordance with the principles of thepresent invention;

FIG. 2 is a block diagram of a system that may be used to implement oneembodiment of a method of deblurring an image in accordance with theprinciples of the present invention;

FIG. 3 is a block diagram of one embodiment of an image deblurringmodule in accordance with the principles of the present invention;

FIGS. 4( a)-(d) is an illustrative example of an implementation of oneembodiment of a rotational convolution method in accordance with theprinciples of the present invention;

FIG. 5 is a flowchart of one embodiment of a method of deblurring animage corrupted by shift variant blurring in accordance with theprinciples of the present invention; and

FIG. 6 is a flowchart of one embodiment of a method of deblurring datacorrupted by shift variant blurring in accordance with the principles ofthe present invention.

DETAILED DESCRIPTION

Referring to FIG. 1 a perspective view of one example of an embodimentof a nuclear imaging assembly 100 in accordance with the principles ofthe present invention is shown. While the illustrated example of anuclear imaging assembly 100 is a SPECT nuclear imaging assembly, theuse of other types of nuclear imaging assemblies are also considered tobe within the scope of the invention. The nuclear imaging assembly 100generally includes a subject support structure 102, a base gantry 104, arotatable gantry 106, and one or more gamma ray imaging devices 108. Asubject 110, such as for example a patient, is typically injected withone or more radiopharmaceuticals or radioisotopes. The injectedradiopharmaceuticals are absorbed by and localize within a target organ,or target in the subject 110. The accumulated radiopharmaceuticals emitenergy in the form of gamma rays or photons that illuminate the target.The nuclear imaging assembly 100 creates images of the distribution ofthe accumulated radioactive pharmaceuticals within the subject target.

The subject support structure 102 supports the subject 110 to be imaged.Examples of subjects 110 include, but are not limited to, patients,animals, portions of animals, and phantoms. The rotatable gantry 106 ismounted on the base gantry 104 and defines a subject receiving aperture112 with a subject imaging region 114 within the subject receivingaperture 112. One or more gamma ray imaging devices 108 are adjustablymounted to the rotatable gantry 106. In one embodiment, the gamma rayimaging devices 108 are positioned at regular intervals around thesubject imaging region 114. For example, a total of three gamma rayimaging devices 108 may be positioned on the rotatable gantry 106 at120° intervals around the subject imaging region 114. In anotherembodiment, the gamma ray imaging devices 108 may be circumferentiallyadjustable to selectively vary their relative spacing with respect toeach other on the rotatable gantry 106. In one embodiment, separatetranslation devices such as motors and drive assemblies (not shown)independently translate the gamma ray imaging devices 108 laterally indirections tangential to the subject imaging region 114 along lineartracks or other appropriate guide structures. In another embodiment, thegamma ray imaging devices 108 are also independently movable in a radialdirection with respect to the subject imaging region 114. In yet anotherembodiment, the gamma ray imaging devices 108 can be selectively cantedor tilted with respect to the radial lines from the center of thesubject imaging region 114. A motor and drive system (not shown) isemployed to control the movement of the gamma ray imaging devices 108.In one embodiment, each gamma ray imaging device 108 can be positionedand controlled individually. In another embodiment, the gamma rayimaging devices 108 can be positioned and controlled together as a unit.

In one embodiment, the base gantry 104 can be advanced towards and/orretracted from the subject support structure 102 so as to appropriatelyposition the subject 110 within the subject imaging region 114 to obtaindesired images of the target. In another embodiment, the subject supportstructure 102 can be advanced towards and/or retracted from the basegantry 104 to achieve the desired positioning of the subject 110 withinthe subject imaging region 114. In yet another embodiment, the subjectsupport structure 102 can be raised or lowered to appropriately positionthe subject 110 within the subject imaging region 114.

Each of the one or more gamma ray imaging devices 108 includes acollimator 116 and a detector 118. Each detector 118 typically includesa scintillation crystal which produces a flash or scintillation of lighteach time it is struck by radiation emanating from the radioactive dyein the subject 110. An array of photomultiplier tubes and associatedcircuitry produces an output signal which is indicative of the (x, y)position of each scintillation on the crystal.

In operation, the subject 110 is placed on the subject support structure102 and the subject support structure 102 is appropriately positionedwithin the subject receiving aperture 112 such that the target ispositioned within the subject imaging region 114. The one or more gammaray imaging devices 108 are appropriately positioned with respect to thetarget to be imaged. The one or more gamma ray detectors 108 are rotatedor indexed in a generally circular orbit about the subject imagingregion 114. The direction of the rotation of the gamma ray imagingdevices 108 defines the axis of rotation. The one or more gamma rayimaging devices 108 detect the radiation emitted by the target from aplurality of different directions and capture multiple two dimensionalimages, where each image provides a different angular view of thetarget. The collected two dimensional images are used to compute orreconstruct three dimensional volumetric representations of the target.

In another embodiment of the invention, one or more gamma ray imagingdevices 108 are mounted onto a base gantry. The subject supportstructure includes a rotatable subject support portion. The one or moregamma ray imaging devices 108 are maintained in stationary positionswith respect to the subject imaging region. In operation, the subject isappropriately positioned on the rotatable subject support portion andthe rotatable subject support portion is appropriately positioned withinthe subject imaging region. The rotatable subject support portion isrotated about an axis of rotation, where the axis of rotation isgenerally parallel to the planar faces of the one or more gamma rayimaging devices 108. The one or more gamma ray imaging devices 108detect the radiation emitted by the target from a plurality of differentdirections and capture multiple two dimensional images, where each imageprovides a different angular view of the target. The collected twodimensional images are used to compute or reconstruct three dimensionalvolumetric representations of the target.

Referring to FIG. 2, a block diagram of a system that may be used toimplement one embodiment of a method of deblurring an image inaccordance with the principles of the present invention is shown.Examples of devices that may include the system 200 include, but are notlimited to, a computer. In one embodiment, the system 200 is integratedas a component of the nuclear imaging assembly 100. In one embodiment,the system 200 generally includes a controller 202 communicativelycoupled to a memory 204, an input device 206, and an output device 208.The controller 202 generally retrieves and executes machine readableinstructions or software programs that are stored in the memory 204.

The memory 204 generally includes an operating system module 210 and adata deblurring module 212. The memory 204 may include additionalapplication modules that may facilitate the operation of the system 200and image deblurring functions. The memory 204 may include one or moreof a non-volatile memory, a volatile memory, and/or one or more storagedevices. Examples of non-volatile memory include, but are not limitedto, electrically erasable programmable read only memory (EEPROM) andread only memory (ROM). Examples of volatile memory include, but are notlimited to, static random access memory (SRAM), and dynamic randomaccess memory (DRAM). Examples of storage devices include, but are notlimited to, hard disk drives, compact disc drives, digital versatiledisc drives, and flash memory devices.

The controller 202 is communicatively coupled to one or more inputdevices 206 and one or more output devices 208. Examples of inputdevices 206 include, but are not limited to, a scanner, a memory storagedevice reader, a keyboard, and a mouse. In one embodiment, one or moreinput/output interfaces are provided to enable data transfer between thecontroller 202 and other devices, such as for example, a nuclear imagingassembly 100 and another computer. In one embodiment, an input/outputinterface is provided to a network that enables the exchange of databetween the nuclear imaging assembly 100 and the system 200. Examples ofoutput devices 208 include, but are not limited to, a display screen, acomputer storage device, and a printer. Examples of computer storagedevices include hard disk drives, compact disc drives, digital versatiledisc drives, and flash memory devices.

Referring to FIG. 3 a block diagram of one embodiment of a datadeblurring module 212 in accordance with the principles of the presentinvention is shown. The two dimensional images of the target collectedby the one or more gamma ray imaging devices 108 are typically corruptedby target specific non-uniform attenuation, collimator specific blurringeffect, and target specific scattering effect. The target specificscattering is also known as the Compton scatter effect. One embodimentof the data deblurring module 212 generally includes a datareconstruction module 302, a point spread function (PSF) determinationmodule 304, a data blurring module 306, and a data filter module 308.The data deblurring module 212 generally receives the corrupted versionof the two dimensional images of the target collected by the one or moregamma ray imaging devices 108 for processing and generates a deblurredversion of a three dimensional image of the target.

Another embodiment of the data deblurring module 212 generally includesa PSF determination module 304, a data blurring module 306, and a datafilter module 308. The data deblurring module 212 generally receives athree dimensional image of the target reconstructed from the corruptedtwo dimensional images for processing. The reconstructed threedimensional image is typically corrupted by shift variant blurring. Thedata deblurring module 212 processes the received reconstructed threedimensional image and generates a deblurred version of the reconstructedthree dimensional image of the target.

In one embodiment, PSF determination module 304, the data blurringmodule 306, the data filter module 308, and the optional the datareconstruction module 302 are all included within a single device. Inone embodiment, the PSF determination module 304, the data blurringmodule 306, the data filter module 308, and the optional datareconstruction module 302 are distributed over one or morecommunicatively coupled devices.

As mentioned previously, the two dimensional images of the targetcollected by the one or more gamma ray imaging devices 108 are typicallycorrupted by target specific non-uniform attenuation, collimatorspecific blurring effect, and target specific scattering effect. The twodimensional images of the target may also be corrupted by uniformattenuation. The data reconstruction module 302 generally receives thetwo dimensional images collected by the one or more gamma ray imagingdevices 108 and generates a reconstructed three dimensional image of thetarget with non-uniform attenuation compensation.

In one embodiment, a transmission scan of the target is provided to thedata reconstruction module 302. The transmission scan of the target istypically acquired using, for example, x-rays or gamma rays. The datareconstruction module 302 uses the transmission scan of the target togenerate an attenuation map of the target. The attenuation map is animage of the linear attenuation coefficients associated with a specifictarget for a specified photon energy level. In one embodiment, anattenuation map of the target is provided directly to the datareconstruction module 302. In one embodiment, a magnetic resonanceimaging (MRI) system is used to generate an attenuation map of thetarget. It should be noted that while the use of a number of differenttechniques for generating an attenuation map of the target have beendescribed, the use of alternative techniques for generating anattenuation map of the target that may be apparent to one of ordinaryskill in the art are also considered to be within the scope of theinvention.

One embodiment of the data reconstruction module 302 employs a filteredbackprojection (FBP) reconstruction algorithm and uses the twodimensional images of the target and the attenuation map of the targetto generate a reconstructed three dimensional image of the target thathas been compensated for the non-uniform attenuation. One embodiment ofthe data reconstruction module 302 employs a filtered backprojection(FBP) reconstruction algorithm to generate a reconstructed threedimensional image of the target that has been compensated for theuniform attenuation. In one embodiment, the data reconstruction module302 uses Novikov's FBP algorithm to generate the reconstructed threedimensional image of the target with non-uniform attenuationcompensation. In one embodiment, the data reconstruction module 302 usesNovikov's FBP algorithm to generate the reconstructed three dimensionalimage of the target with uniform attenuation compensation. Oneembodiment of the data reconstruction module 302 employs an iterativereconstruction algorithm and uses the two dimensional images of thetarget and the attenuation map of the target to generate a reconstructedthree dimensional image of the target that has been compensated for thenon-uniform attenuation. One embodiment of the data reconstructionmodule 302 employs an iterative reconstruction algorithm to generate areconstructed three dimensional image of the target that has beencompensated for the uniform attenuation. It should be noted that whilethe use of techniques for compensating non-uniform and/or uniformattenuation in the reconstruction of a three dimensional image has beendescribed, the use of alternative techniques for compensatingnon-uniform and/or uniform attenuation that may be known to one ofordinary skill in the art may be used without departing from the spiritof the invention.

While the reconstructed three dimensional image generated by the datareconstruction module 302 has been compensated for the non-uniformattenuation, the reconstructed three dimensional image remains corruptedby the collimator specific blurring effect, and the Compton scattereffect. The combined effects of the collimator specific blurring and theCompton scatter effect results in the presence of shift variant ornon-stationary blurring in the reconstructed three dimensional image.The shift variant blurring can be characterized using a shift variantposition spread function (PSF) in the image domain.

In one embodiment, the PSF determination module 304 receives thereconstructed three dimensional image generated by the datareconstruction module 302 and generates a shift variant PSFcharacterizing the shift variant blurring present in the reconstructedthree dimensional image in the image domain. In one embodiment, the PSFdetermination module 304 generates a shift variant PSF characterizingthe shift variant blurring attributable to the combination of thecollimator specific blurring and the Compton scatter effect present inthe reconstructed three dimensional image in the image domain. In oneembodiment, the PSF determination module 304 generates a shift variantPSF characterizing the shift variant blurring attributable to thecollimator blurring effect present in the reconstructed threedimensional image in the image domain. In one embodiment, the PSFdetermination module 304 generates a shift variant PSF characterizingthe shift variant blurring attributable to the Compton scatter effectpresent in the reconstructed three dimensional image in the imagedomain.

In one embodiment, the PSF determination module 304 determines a firstshift variant PSF or a collimator specific PSF characterizing collimatorspecific blurring in the reconstructed three dimensional image. Thefirst shift variant PSF of the reconstructed three dimensional image isin the image domain. In one embodiment, the PSF determination module 304determines the first shift variant PSF based on collimatorspecifications provided by the collimator manufacturer. In oneembodiment, the PSF determination module 304 derives the first shiftvariant PSF based on an analysis of collimator generated projections ofa point source and/or line source. The PSF determination module 304determines a shift variant PSF for each of the two dimensionalprojections generated by the collimator and uses the shift variant PSFsfor the two dimensional projections to derive the first shift variantPSF of the reconstructed three dimensional image in the image domain. Inone embodiment, The PSF determination module 304 employs Novikov's FBPalgorithm to derive the first shift variant PSF of reconstructed threedimensional image in the image domain from the detector PSF where thedetector is associated with the collimator. While a number of differenttechniques have been described for determining the first shift variantPSF associated with collimator specific blurring for a reconstructedthree dimensional image in the image domain, alternative techniques fordetermining the first shift variant PSF of the reconstructed threedimensional image in the image domain known to one of ordinary skill inthe art may be used without departing from the spirit of the invention.

In one embodiment, the PSF determination module 304 determines a secondshift variant PSF or a target specific scattering PSF characterizing theCompton scatter effect on the reconstructed three dimensional image. Thesecond shift variant PSF of the reconstructed three dimensional image isin the image domain. In one embodiment, the PSF determination module 304determines the second shift variant PSF based on an attenuation map ofthe target.

In one embodiment, an attenuation map of the target is provided directlyto the PSF determination module 304. In one embodiment, the attenuationmap is provided from the data reconstruction module 302 to the PSFdetermination module 304. In one embodiment, a transmission scan of thetarget is provided to the PSF determination module 304. The PSFdetermination module 304 uses the transmission scan to generate anattenuation map of the target. In one embodiment, a magnetic resonanceimaging (MRI) system is used to generate an attenuation map of thetarget. It should be noted that while the use of a number of differenttechniques for generating an attenuation map of the target have beendescribed, the use of alternative techniques for generating anattenuation map and determining the scatter effect of the target thatmay be apparent to one of ordinary skill in the art are also consideredto be within the scope of the invention.

While one technique has been described for determining the second shiftvariant PSF associated with the Compton scatter effect for areconstructed three dimensional image in the image domain, alternativetechniques for determining the second shift variant PSF of thereconstructed three dimensional image in the image domain known to oneof ordinary skill in the art may be used without departing from thespirit of the invention.

In one embodiment, the PSF determination module 304 uses the first shiftvariant PSF associated with collimator specific blurring and the secondshift variant PSF associated with the Compton scatter effect to derivethe shift variant PSF characterizing the shift variant blurring presentin the reconstructed three dimensional image in the image domain. Whileone manner of determining a shift variant PSF for the reconstructedimage in the image domain characterizing the shift variant blurringattributable to the combination of the collimator specific blurring andthe Compton scatter effect present in the reconstructed threedimensional image has been described, the use of alternative techniquesfor determining the shift variant PSF for the reconstructed image in theimage domain are also considered to be within the scope of theinvention.

For illustrative purposes, the following definitions will be used. Thefunction g(x, y, z) will be used to represent the reconstructed threedimensional image generated by the data reconstruction module 302. Thefunction h(x, y, z; x₀, y₀, z₀) will be used to represent the shiftvariant PSF representative of the shift variant blurring present in thereconstructed three dimensional image generated by the PSF generationmodule 304. The function f(x, y, z) will be used to represent anunblurred or true version of the reconstructed three dimensional image.

The relationship between the reconstructed three dimensional image g(x,y, the shift variant PSF h(x, y, z; x₀, y₀, z₀), and the unblurredversion of the reconstructed three dimensional image f(x, y, z) at apoint (x₀, y₀, z₀) of the three dimensional image of the target can bemathematically modeled in the spatial domain as indicated in Equation(1) below:

$\begin{matrix}{{g\left( {x_{0},y_{0},z_{0}} \right)} = {\sum\limits_{x,y,z}{{f\left( {x,y,z} \right)}{h\left( {{x - x_{0}},{y - y_{0}},{{z - z_{0}};x_{0}},y_{0},z_{0}} \right)}}}} & {{Equation}\mspace{14mu} (1)}\end{matrix}$

The data blurring module 306 generally receives the reconstructed threedimensional image g(x, y, z) and the shift variant PSF h(x, y, z, x₀,y₀, z₀) characterizing the shift variant blurring present in thereconstructed three dimensional image g(x, y, z) and generates ablurrier version of the reconstructed three dimensional image having atarget shift invariant or stationary PSF. One embodiment of the datablurring module 306 selects a target shift invariant PSF. In oneembodiment, the selected target shift invariant PSF is the PSF of theblurriest section of the reconstructed three dimensional image g(x, y,z). In one embodiment, the selected target shift invariant PSF is ashift invariant PSF that exceeds the PSF of the blurriest portion of thereconstructed three dimensional image. In one embodiment, the selectedtarget PSF is the PSF of a relatively blurrier section of thereconstructed three dimensional image. The data blurring module 306derives a blurring kernel. The derived blurring kernel has a shiftvariant PSF, such that the combination of the shift variant PSF h(x, y,z; x₀, y₀, z₀) of the reconstructed three dimensional image and theshift variant PSF of the blurring kernel generates the target shiftinvariant PSF.

For illustrative purposes the function h₀(x, y, z) will be used torepresent the selected target shift invariant PSF, the function k(x, y,z; x₀, y₀, z₀) will be used to represent the blurring kernel defined bythe shift variant PSF. The relationship between the shift variant PSFh(x, y, z; x₀, y₀, z₀) of the reconstructed three dimensional image, theselected target shift invariant PSF h₀(x, y, z), and the blurring kerneldefined by the shift variant PSF k(x, y, z; x₀, y₀, z₀) can bemathematically modeled in the spatial domain as indicated in Equation(2) below:

h ₀(x,y,z)=h(x,y,z;x ₀ ,y ₀ ,z ₀)*k(x,y,z;x ₀ ,y ₀ ,z ₀)  Equation (2)

where the symbol “*” represents a three dimensional convolutionoperator.

For illustrative purposes, H(u, v, w; x₀, y₀, z₀) will be used torepresent the Fourier transform of h(x, y, z; x₀, y₀, z₀), H₀(u, v, w)will be used to represent the Fourier transform of h₀(x, y, z), and K(u,v, w; x₀, y₀, z₀) will be used to represent the Fourier transform ofk(x, y, z; x₀, y₀, z₀). The parameters u, v, and w represent frequenciesalong the x, y, and z axes, respectively. Based on the relationshipbetween the functions h₀(x, y, z), h(x, y, z; x₀, y₀, z₀), and k(x, y,z; x₀, y₀, z₀) defined in the spatial domain in Equation (2), therelationship between H(u, v, w; x₀, y₀, z₀), H₀(u, v, w), and K(u, v, w;x₀, y₀, z₀) can be mathematically modeled in the frequency domain asindicated in Equation (3) below:

H ₀(u,v,w)=H(u,v,w;x ₀ ,y ₀ ,z ₀)×K(u,v,w;x ₀ ,y ₀ ,z ₀)  Equation (3)

Dividing both sides of the Equation (3) by H(u, v, w; x₀, y₀, z₀),generates the relationship indicated in Equation (4) below:

K(u,v,w,x ₀ ,y ₀ ,z ₀)=H ₀(u,v,w)/H(u,v,w,x ₀ ,y ₀ ,z ₀)  Equation (4)

In one embodiment, the data blurring module 306 receives thereconstructed three dimensional image g(x, y, z) and the shift variantPSF h(x, y, z; x₀, y₀, z₀), characterizing the shift variant blurringpresent in the reconstructed three dimensional image. The data blurringmodule 306 selects the PSF of the blurriest section of the reconstructedthree dimensional image g(x, y, z) as the target shift invariant PSFh₀(x, y, z). The data blurring module 306 determines the Fouriertransforms H₀(u, v, w) and H(u, v, w; x₀, y₀, z₀) of the target shiftinvariant PSF h₀(x, y, z) and the shift variant PSF h(x, y, z; x₀, y₀,z₀), respectively. The data blurring module 306 divides the Fouriertransform of the target shift invariant PSF H₀(u, v, w) by the Fouriertransform of the shift variant PSF H(u, v, w; x₀, y₀, z₀) of the threedimensional reconstructed image to determine the Fourier transform ofthe shift variant PSF defining the blurring kernel K(u, v, w; x₀, y₀,z₀). The data blurring module 306 takes the inverse Fourier transform ofK(u, v, w; x₀, y₀, z₀) to derive the shift variant PSF function definingthe blurring kernel k(x, y, z; x₀, y₀, z₀). The data blurring module 306applies the derived blurring kernel k(x, y, z, x₀, y₀, z₀) to thereconstructed three dimensional image g(x, y, z) having the shiftvariant PSF of h(x, y, z; x₀, y₀, z₀) thereby generating the blurrierversion of the reconstructed three dimensional image having the targetshift invariant PSF h₀(x, y, z).

For illustrative purposes the function q(x, y, z) will be used torepresent the blurrier version of the reconstructed three dimensionalimage having the target shift invariant PSF h₀(x, y, z). Therelationship between the blurrier version of the three dimensionalreconstructed image q(x, y, z) having the shift invariant PSF h₀(x, y,z), the reconstructed three dimensional image g(x, y, z) having theshift variant PSF h(x, y, z; x₀, y₀, z₀), and the blurring kernel k(x,y, z; x₀, y₀, z₀) at a point (x₀, y₀, z₀) of the reconstructed threedimensional image can be mathematically modeled as indicated in Equation(5) below:

$\begin{matrix}{{q\left( {x_{0},y_{0},z_{0}} \right)} = {\sum\limits_{x,y,z}{{g\left( {x,y,z} \right)}{k\left( {{x - x_{0}},{y - y_{0}},{{z - z_{0}};x_{0}},y_{0},z_{0}} \right)}}}} & {{Equation}\mspace{14mu} (5)}\end{matrix}$

The data blurring module 304 forwards the blurrier version of thereconstructed three dimensional image q(x, y, z) having the target shiftinvariant PSF h₀(x, y, z) to the data filter module 308 for processing.

One embodiment of the data blurring module 306 derives the shift variantPSF defining the blurring kernel k(x, y, z, x₀, y₀, z₀) based on arelationship between the variances of the target shift invariant PSFh₀(x, y, z), and the shift variant PSF h(x, y, z; x₀, y₀, z₀)characterizing the shift variant blurring present in the reconstructedthree dimensional image g(x, y, z), and the PSF defining the blurringkernel k(x, y, z x₀, y₀, z₀). For illustrative purposes, the variancesof the PSF functions h₀(x, y, z), h(x, y, z, x₀, y₀, z₀), and k(x, y, z;x₀, y₀, z₀) will be represented as σ_(h0) ², σ² _(h), and σ² _(k),respectively. The relationship between the variances. σ_(h0) ², σ² _(h),and σ² _(k) can be mathematically modeled using a Gaussian model asindicated in Equation (6) below:

σ_(h0) ²=σ² _(h)+σ² _(k)  Equation (6)

Subtracting σ² _(h) from both sides of the Equation (6) generates therelationship indicated in Equation (7) below:

σ² _(k)=σ_(h0) ²−σ² _(h)  Equation (7)

The data blurring module 306 receives the reconstructed threedimensional image g(x, y, z) and the shift variant PSF h(x, y, z; x₀,y₀, z₀), characterizing the shift variant blurring present in thereconstructed three dimensional image. The data blurring module 306selects the PSF of the blurriest section of the reconstructed threedimensional image g(x, y, z) as the target shift invariant PSF h₀(x, y,z). The data blurring module 306 derives the variances σ_(h0) ², and σ²_(h) of the PSF functions h₀(x, y, z) and h(x, y, z, x₀, y₀, z₀),respectively. The data blurring module 306 subtracts the variance of theshift invariant PSF characterizing the shift variant blur present in thereconstructed three dimensional image σ² _(h) from the variance of thetarget shift invariant PSF σ_(h0) ² to determine the variance of theshift variant PSF defining the blurring kernel σ² _(k). The datablurring module 306 uses the variance of the shift variant PSF definingthe blurring kernel σ² _(k) to derive the shift variant PSF defining theblurring kernel k(x, y, z; x₀, y₀, z₀). The data blurring module 306applies the derived blurring kernel k(x, y, z; x₀, y₀, z₀) to thereconstructed three dimensional image g(x, y, z) having the shiftvariant PSF of h(x, y, z; x₀, y₀, z₀) thereby generating the blurrierversion of the reconstructed three dimensional image q(x, y, z) havingthe target shift invariant PSF h₀(x, y, z). The data blurring module 304forwards the blurrier version of the reconstructed three dimensionalimage q(x, y, having the target shift invariant PSF h₀(x, y, z) to thedata filter module 308 for processing.

In one embodiment, the data blurring module 306 uses a rotationalconvolution method to further blur the reconstructed three dimensionalimage g(x, y, z) to generate the blurrier version of the reconstructedthree dimensional image q(x, y, z) having the target shift invariant PSFh₀(x, y, z). The rotational convolution method can be used in instanceswhere the shift variant PSF h(x, y, z; x₀, y₀, z₀) primarily reflectsthe shift variant collimator specific blurring in the reconstructedthree dimensional image and where the two dimensional images of thetarget were captured using one or more gamma ray imaging devices 108that were rotated in generally circular orbit around the target.

An illustrative example of the implementation of the rotationalconvolution method in accordance with the principles of the presentinvention is described with reference to FIGS. 4( a)-(d). FIG. 4( a)depicts an example of a true unblurred three dimensional image of atarget f(x, y, z) consisting of three spheres. The origin of thecoordinate system coincides with the axis of rotation of the gamma rayimaging devices 108.

FIG. 4( b) illustrates an example of a reconstructed three dimensionalimage g(x, y, z) of the three spheres having shift invariant blurcharacterized by a shift invariant PSF h(x, y, z; x₀, y₀, z₀). The widthof the shift variant PSF h(x, y, z; x₀, y₀, z₀) is generally stationaryalong the radial direction (along the x-axis of the coordinate system).The width of the shift variant PSF h(x, y, z; x₀, y₀, z₀) becomesincreasingly narrower in the tangential direction (along the y-axis ofthe coordinate system) as the distance from the axis of rotation (originof the coordinate system) increases. The reconstructed three dimensionalimage g(x, y, z) of the sphere disposed at the origin of the coordinatesystem represents the blurriest section of the reconstructed threedimensional image and is selected by the data blurring module 308 as thetarget shift invariant blurring PSF h₀(x, y, z).

Referring to the example illustration in FIG. 4( c), the data blurringmodule 308 incrementally rotates the reconstructed three dimensionalimage g(x, y, z) about the origin of the coordinate system through anumber of relatively small angles in a clockwise direction and a througha number of relatively small angles in a counter-clockwise direction.The blurriest section of the reconstructed three dimensional image g(x,y, z) define the boundaries of the rotation of the reconstructed threedimensional image g(x, y, z).

The data blurring module 308 determines a weighted sum of the rotatedversions of the reconstructed three dimensional image g(x, y, z) therebygenerating a blurrier version of the reconstructed three dimensionalimage q(x, y, z) characterized by the target shift invariant blurringPSF h₀(x, y, z). This relationship can be mathematically modeled asindicated in Equation (8) below:

$\begin{matrix}{{q\left( {x_{0},y_{0},z_{0}} \right)} = {\sum\limits_{n}{a_{n}{g\left( {x_{0},y_{0},z_{0}} \right)}_{n\; \Delta}}}} & {{Equation}\mspace{14mu} (8)}\end{matrix}$

where a_(n) represents the weighing factors and g_(Δ) represents theincrementally rotated image resulting from each incremental angularrotations of the reconstructed three dimensional version of the imageg(x, y, z). The weighted factors a_(n) are based on the distance of thepoint (x₀, y₀, z₀) from the origin of the coordinate system or the axisof rotation. Referring to FIG. 4( d), an example of a blurrier versionof the reconstructed three dimensional image q(x, y, z) having thetarget shift invariant PSF of h₀(x, y, z) is shown. Note that thespheres that are further removed from the origin or the axis of rotationare now as blurred as the sphere disposed at the origin of thecoordinate system. The data blurring module 304 forwards the blurrierversion of the reconstructed three dimensional image q(x, y, z) havingthe target shift invariant PSF h₀(x, y, z) to the data filter module 308for processing.

The data filter module 308 generally receives the blurrier version ofthe reconstructed three dimensional image q(x, y, z) having shiftinvariant blur characterized by the target shift invariant PSF h₀(x, y,z) and generates a deblurred or true version of the reconstructed threedimensional image f(x, y, z). The data filter module 308 applies a shiftinvariant linear filter to the blurrier version of the reconstructedthree dimensional image q(x, y, z) to compensate for the target shiftinvariant PSF h₀(x, y, z). In one embodiment, the data filter module 308applies a non-iterative filter to the blurrier version of thereconstructed three dimensional image q(x, y, z) to generate thedeblurred version of the reconstructed three dimensional image f(x, y,z).

In one embodiment, the data filter module 308 de-convolves the blurrierversion of the reconstructed three dimensional image q(x, y, z) tocompensate for the target shift invariant PSF h₀(x, y, z). In oneembodiment, the data filter module 308 implements a convolutionprocedure in the image domain to deblur the blurrier version of thereconstructed three dimensional image q(x, y, z). In one embodiment, thedata filter module 308 implements a multiplication procedure in thefrequency domain to deblur the blurrier version of the reconstructedthree dimensional image q(x, y, z). In one embodiment, the data filtermodule 308 employs a fast Fourier domain de-blurring filter to deblurthe blurrier version of the reconstructed three dimensional image q(x,y, z). In one embodiment, the data filter module 308 applies aniterative filter to the blurrier version of the reconstructed threedimensional image q(x, y, z) to generate the deblurred version of thereconstructed three dimensional image f(x, y, z). While a number ofdifferent types of filters that can be used to compensate for the targetshift invariant PSF in the blurrier version of the reconstructed threedimensional image have been described, the use of alternative filters tocompensate for shift invariant PSF may be used without departing fromthe spirit of the invention.

It should be noted that while systems implemented using software orfirmware executed by hardware have been described above, those havingordinary skill in the art will readily recognize that the disclosedsystems could be implemented exclusively in hardware through the use ofone or more custom circuits, such as for example, application-specificintegrated circuits (ASICs) or any other suitable combination ofhardware and/or software.

Referring to FIG. 5 a flowchart of one embodiment of a method 500 ofdeblurring an image in accordance with the principles of the presentinvention is shown. A set of two dimensional images of a targetcollected by a nuclear imaging assembly 100 is received by the datareconstruction module 302 at step 502. The two dimensional images arealso referred to as two dimensional projection images. The received twodimensional images of the target are typically corrupted by targetspecific non-uniform attenuation, collimator specific blurring effect,and target specific scattering effect. The data reconstruction module302 reconstructs a three dimensional image of the target from thereceived two dimensional images of the target using a filteredbackprojection algorithm (FBP) at step 504. In another embodiment, thedata reconstruction module 302 reconstructs a three dimensional image ofthe target from the received two dimensional images of the target usingan iterative algorithm. The reconstructed three dimensional imagegenerated by the data reconstruction module 302 has been compensated forthe non-uniform attenuation. The reconstructed three dimensional imagegenerated by the data reconstruction module 302 has also beencompensated for the uniform attenuation. However, the reconstructedthree dimensional image remains corrupted by the collimator specificblurring effect, and the target specific scattering effect.

The PSF determination module 304 generates a shift variant PSFcharacterizing the shift variant blurring present in the reconstructedthree dimensional image in the image domain at step 506. The shiftvariant PSF of the reconstructed three dimensional image characterizesthe shift variant blurring attributable to the combination of thecollimator specific blurring and the target specific scatter effectpresent in the reconstructed three dimensional image in the imagedomain.

The data blurring module 306 selects the PSF of the blurriest portion ofthe reconstructed three dimensional image as a target shift invariantPSF at step 508. The data blurring module 306 derives a blurring kerneldefined by a shift variant PSF, such that the combination of the shiftvariant PSF of the reconstructed three dimensional image and the shiftvariant PSF of the blurring kernel generate the target shift invariantPSF at step 510. The data blurring module 308 applies the derivedblurring kernel to the reconstructed three dimensional image therebygenerating a blurrier version of the reconstructed three dimensionalimage at step 512. The blurrier version of the reconstructed threedimensional image has shift invariant blurring characterized by thetarget shift invariant PSF. The data filter module 308 applies a shiftinvariant linear filter to the blurrier version of the reconstructedthree dimensional image having the invariant target PSF therebygenerating a deblurred version of the reconstructed three dimensionalimage at step 514.

It should be noted that while the steps in the method 500 have beendescribed in a particular order, performing one or more of the steps ina different order, or performing a subset of the described steps arealso considered to be within the scope of the invention.

Referring to FIG. 6, a flowchart of a method 600 of deblurring datacorrupted by shift variant blurring in accordance with the principles ofthe present invention is shown. A first version of data having shiftvariant blurring characterized by a first shift variant point spreadfunction is provided at step 602. A target shift invariant point spreadfunction is selected at step 604. A second shift variant point spreadfunction is derived at step 606, wherein a combination of the first andsecond shift variant point spread functions generates the target shiftinvariant point spread function. The second shift variant point spreadfunction is applied to the first version of the data thereby generatinga second version of the data having shift invariant blurringcharacterized by the target shift invariant point spread function atstep 608. A linear shift invariant filter is applied to the secondversion of the data thereby generating a deblurred version of the data.It should be noted that while the steps in the method 600 have beendescribed in a particular order, performing one or more of the steps ina different order is also considered to be within the scope of theinvention.

One embodiment, a machine readable medium stores a machine executableprogram for deblurring data corrupted by shift variant blurring. Themachine readable medium includes machine readable code for providing afirst version of data having shift variant blurring characterized by afirst shift variant point spread function, machine readable code forselecting a target shift invariant point spread function, machinereadable code for deriving a second shift variant point spread functionwherein a combination of the first and second shift variant point spreadfunctions generates the target shift invariant point spread function,machine readable code for applying the second shift variant point spreadfunction to the first version of the data thereby generating a secondversion of the data having shift invariant blurring characterized by thetarget shift invariant point spread function, and machine readable codefor applying a linear shift invariant filter to the second version ofthe data thereby generating a deblurred version of the data.

One embodiment of a method for deblurring data corrupted by shiftvariant blurring includes providing a first version of data having shiftvariant blurring, blurring the first version of the data further using ashift variant blurring kernel thereby generating a second version ofdata having shift invariant blurring, and applying a linear shiftinvariant filter to the second version of the data thereby generating adeblurred version of the data.

It should be noted that while systems and methods for deblurring a threedimensional image corrupted by a shift variant PSF has been described,systems and methods for deblurring one dimensional data as well assystems and methods for deblurring two dimensional data, such as forexample, two dimensional images are also considered to be within thescope of the invention. For example, in geophysics, the migration ofseismic data is often modeled as shift variant convolution. The systemsand methods for deblurring data corrupted by shift variant PSF may beused to deblur such data. Two dimensional ultrasound images are anexample of two dimensional data that may be corrupted by shift variantblurring. The methods and systems for deblurred data corrupted by shiftvariant blurring may be used to compensate for the shift variantblurring in such two dimensional data. While examples of specific oneand two dimensional data that can be deblurred using the disclosedsystems and methods for deblurring data corrupted by shift variantblurring, the processing of other types of one and two dimensional databy the disclosed systems and methods for deblurring data corrupted byshift variant blurring are also considered to be within the scope of theinvention.

While the embodiments of the invention disclosed herein are presentlyconsidered to be preferred, various changes, and modifications can bemade without departing from the spirit and scope of the invention. Thescope of the invention is indicated in the appended claims, and allchanges that come within the meaning and range of equivalents areintended to be embraced therein.

What is claimed is:
 1. A method of deblurring data corrupted by shiftvariant blurring, the method comprising: providing a first version ofdata having shift variant blurring characterized by a first shiftvariant point spread function; applying a second shift variant pointspread function to the first version of the data, thereby generating asecond version of the data, wherein the second version of the data is ablurrier version of the data than is the first version of the data; andapplying a linear shift invariant filter to the second version of thedata to generate a deblurred version of the data.
 2. The method of claim1, wherein the first version of data comprises a three dimensionalimage, the method further comprising: determining a collimator specificpoint spread function characterizing a first component of the shiftvariant blurring attributable to a collimator specific blurring effectin an image domain; determining a target specific scattering pointspread function characterizing a second component of the shift variantblurring attributable to a target specific scattering effect in an imagedomain; and deriving the first variant point spread function based onthe collimator specific point spread function and the target specificpoint spread function.
 3. The method of claim 2, wherein determining acollimator specific point spread function in the image domain comprisesdetermining the collimator specific point spread function based oncollimator data selected from a group consisting of manufacturersupplied collimator specification data, collimator projection data, andcollimator detector point spread function data.
 4. The method of claim2, wherein determining a collimator specific point spread function inthe image domain comprises determining a collimator specific pointspread function using a reconstruction algorithm selected from a groupconsisting of a filtered backprojection (FBP) reconstruction algorithmand an iterative reconstruction algorithm.
 5. The method of claim 1,further comprising: selecting a target shift invariant point spreadfunction; and deriving the second shift variant point spread function,wherein a combination of the first and second shift variant point spreadfunctions generates the target shift invariant point spread function. 6.The method of claim 5, wherein selecting a target shift invariant pointspread function comprises selecting a target shift invariant pointspread function from a group consisting of a point spread function of ablurriest section of the first version of the data, a point spreadfunction that exceeds a point spread function of blurriest section ofthe first version of the data, and a point spread function of arelatively blurry section of the first version of the data.
 7. Themethod of claim 5, wherein deriving the second shift variant pointspread function comprises: determining a Fourier transform of the targetshift invariant point spread function; determining a Fourier transformof the first shift variant point spread function; generating a Fouriertransform of the second shift variant point spread function by dividingthe Fourier transform of the target shift invariant point spreadfunction by the Fourier transform of the first shift variant pointspread function; and determining the inverse Fourier transform of thesecond shift variant point spread function thereby generating the secondshift variant point spread function.
 8. The method of claim 1, whereinderiving the second shift variant point spread function comprises:determining a variance of the target shift invariant point spreadfunction; determining a variance of the first shift variant point spreadfunction; generating a variance of the second shift variant point spreadfunction by subtracting the variance of the first shift variant pointspread function from the variance of the target shift invariant pointspread function; and deriving the second shift variant point spreadfunction from the variance of the second shift variant point spreadfunction.
 9. The method of claim 1, wherein applying the second shiftvariant point spread function to the first version of the data comprisesapplying the second shift variant point spread function to the firstversion of the data using a rotational convolution method.
 10. Themethod of claim 1, wherein applying a linear shift invariant filter tothe second version of the data comprises applying a linear shiftinvariant filter selected from a group consisting of a non-iterativefilter, an iterative filter, a convolution procedure based filter, afrequency domain multiplication procedure based filter, and a fastFourier domain de-blurring filter.
 11. A machine readable medium forstoring a machine executable program for deblurring data corrupted byshift variant blurring, comprising: machine readable code for providinga first version of data having shift variant blurring characterized by afirst shift variant point spread function; machine readable code forapplying a second shift variant point spread function to the firstversion of the data thereby generating a second version of the data,wherein the second version of the data is a blurrier version of the datathan the first version of the data; and machine readable code forapplying a linear shift invariant filter to the second version of thedata thereby generating a deblurred version of the data.
 12. The machinereadable medium of claim 15, wherein the first version of data comprisesa three dimensional image, the machine readable medium furthercomprising: machine readable code for determining a collimator specificpoint spread function characterizing a first component of the shiftvariant blurring attributable to a collimator specific blurring effectin an image domain; machine readable code for determining a targetspecific scattering point spread function characterizing a secondcomponent of the shift variant blurring attributable to a targetspecific scattering effect in an image domain; and machine readable codefor deriving the first variant point spread function based on thecollimator specific point spread function and the target specific pointspread function.
 13. The machine readable medium of claim 12, whereinthe machine readable code for determining a collimator specific pointspread function in the image domain comprises machine readable code fordetermining the collimator specific point spread function based oncollimator data selected from a group consisting of manufacturersupplied collimator specification data, collimator projection data, andcollimator detector point spread function data.
 14. The machine readablemedium of claim 12, wherein the machine readable code for determining acollimator specific point spread function in the image domain comprisesmachine readable code for determining a collimator specific point spreadfunction using Novikov's filtered backprojection (FBP) algorithm. 15.The machine readable medium, further comprising: machine readable codefor selecting a target shift invariant point spread function; andmachine readable code for deriving the second shift variant point spreadfunction, wherein a combination of the first and second shift variantpoint spread functions generates the target shift invariant point spreadfunction;
 16. The machine readable medium of claim 15, wherein themachine readable code for selecting a target shift invariant pointspread function comprises machine readable code for selecting a targetshift invariant point spread function from a group consisting of a pointspread function of a blurriest section of the first version of the data,a point spread function that exceeds a point spread function of ablurriest section of the first version of the data, and a point spreadfunction of a relatively blurry section of the first version of thedata.
 17. The machine readable medium of claim 15, wherein the machinereadable code for deriving the second shift variant point spreadfunction comprises: machine readable code for determining a Fouriertransform of the target shift invariant point spread function; machinereadable code for determining Fourier transform of the first shiftvariant point spread function; machine readable code for generating aFourier transform of the second shift variant point spread function bydividing the Fourier transform of the target shift invariant pointspread function by the Fourier transform of the first shift variantpoint spread function; and machine readable code for determining theinverse Fourier transform of the second shift variant point spreadfunction thereby generating the second shift variant point spreadfunction.
 18. The machine readable medium of claim 15, wherein themachine readable code for deriving the second shift variant point spreadfunction comprises: machine readable code for determining a variance ofthe target shift invariant point spread function; machine readable codefor determining a variance of the first shift variant point spreadfunction; machine readable code for generating a variance of the secondshift variant point spread function by subtracting the variance of thefirst shift variant point spread function from the variance of thetarget shift invariant point spread function; and machine readable codefor deriving the second shift variant point spread function from thevariance of the second shift variant point spread function.
 19. Themachine readable medium of claim 11, wherein the machine readable codefor applying the second shift variant point spread function to the firstversion of the data comprises machine readable code for applying thesecond shift variant point spread function to the first version of thedata using a rotational convolution method.
 20. The machine readablemedium of claim 11, wherein the machine readable code for applying alinear shift invariant filter to the second version of the datacomprises machine readable code for applying a linear shift invariantfilter selected from a group consisting of a non-iterative filter, aniterative filter, a convolution procedure based filter, a frequencydomain multiplication procedure based filter, and a fast Fourier domainde-blurring filter.